# The open smooth case

Let ${\displaystyle U}$ be a smooth variety and ${\displaystyle X\supseteq U}$ a smooth compactification, such that ${\displaystyle X-U=D}$ is a (smooth) normal crossing divisor. Consider the long exact sequence of relative cohomology groups

${\displaystyle \cdots \longrightarrow H^{i}(X,U)\longrightarrow H^{i}(X)\longrightarrow H^{i}(U)\longrightarrow H^{i+1}(X,U)\longrightarrow \cdots }$
By excision and by Thom isomorphism, we have
${\displaystyle H^{i}(X,U)\simeq H^{i}({\mathcal {N}}_{D},{\mathcal {N}}_{D}-D)\simeq H^{i-2}(D),}$
where ${\displaystyle {\mathcal {N}}_{D}}$ is the normal bundle of ${\displaystyle D}$ in ${\displaystyle X}$. Thus, we have that ${\displaystyle H^{i}(X,U)}$ can be seen as a pure Hodge structure but of the wrong weight, since we would like
${\displaystyle H^{i-2}(D)\longrightarrow H^{i}(X)}$
to be a morphism of pure Hodge structures (of bidegree ${\displaystyle (0,0)}$) but they have weights ${\displaystyle i-2}$ and ${\displaystyle i}$. Thus, we take the Tate twist so that our long exact sequence looks like
${\displaystyle \cdots \longrightarrow H^{i-2}(D)(-1)\longrightarrow H^{i}(X)\longrightarrow H^{i}(U)\longrightarrow H^{i-1}(D)(-1)\longrightarrow \cdots .}$
${\displaystyle H^{i}(U)}$ admits a mixed Hodge structure with induced weight filtration
${\displaystyle W^{k}(H^{i}(U))={\begin{cases}0&k
Where does the decreasing filtration arise? In the classical case, i.e. for a complex manifold ${\displaystyle X}$, we consider the complex of sheaves on ${\displaystyle X}$ of holomorphic ${\displaystyle \bullet }$-forms ${\displaystyle \Omega _{X}^{\bullet }}$ and its naive filtration
${\displaystyle F^{p}\Omega _{X}^{\bullet }:=q\Omega _{X}^{\geq p}=0\to \cdots 0\to \Omega _{X}^{p}\to \Omega _{X}^{p+1}\to \cdots .}$
It is easy to check that induces a decreasing filtration on the hypercohomology of ${\displaystyle \Omega _{X}^{\bullet }}$, which is defined as
${\displaystyle F^{p}\mathbb {H} ^{n}(X,\Omega _{X}^{\bullet }):=q\operatorname {im} (\mathbb {H} ^{n}(X,F^{p}\Omega _{X}^{\bullet })\to \mathbb {H} ^{n}(X,\Omega _{X}^{\bullet })).}$
Since ${\displaystyle \Omega _{X}^{\bullet }}$ is a resolution of the locally constant sheaf of stalk ${\displaystyle \mathbb {C} }$ over ${\displaystyle X}$, we have
${\displaystyle \mathbb {H} ^{n}(X,\Omega _{X}^{\bullet })\simeq H^{n}(X,\mathbb {C} ),}$
which induces a decreasing filtration on ${\displaystyle H^{n}(X,\mathbb {C} )}$. In the case of compact Kähler manifolds, this is what we are looking for and, indeed, the filtration leads us to the proof of (a weaker version of) the Hodge decomposition. Unfortunately, if we try to use the same construction for ${\displaystyle U}$, in general we get no extra information. For instance, for ${\displaystyle U}$ affine,we would get
${\displaystyle F^{n}H^{n}(U,\mathbb {C} )=H^{n}(U,\mathbb {C} ).}$
We need to replace our complex of sheaves ${\displaystyle \Omega _{X}^{\bullet }}$ with something clever. It turns out that if we allow logarithmic singularities along ${\displaystyle D}$, we are able to induce a (non-trivial) decreasing filtration on ${\displaystyle H^{i}(U,\mathbb {C} )}$ from the naive one on the new complex. More precisely, let ${\displaystyle \Omega _{X}^{\bullet }(\operatorname {log} D)}$ be the subcomplex of ${\displaystyle j_{*}\Omega _{U}^{\bullet }}$, where ${\displaystyle j:U\hookrightarrow X}$ is the inclusion of ${\displaystyle U}$ in its compactification such that

• a meromorphic differential ${\displaystyle k}$-form ${\displaystyle \alpha }$ on an open ${\displaystyle V\subset X}$, holomorphic on ${\displaystyle V\cap U}$ is an element of ${\displaystyle \Omega _{X}^{k}(\operatorname {log} D)|_{V}}$ if ${\displaystyle \alpha }$ and ${\displaystyle d\alpha }$ admit poles of order at most 1 in ${\displaystyle V\cap D}$.

We can give an explicit local description of the elements of ${\displaystyle \Omega _{X}^{k}(\operatorname {log} D)}$: Let ${\displaystyle z_{1},\ldots ,z_{n}}$ be local coordinates on an open set ${\displaystyle V}$ of ${\displaystyle X}$, in which ${\displaystyle V\cap D}$ is defined by the equation

${\displaystyle z_{1}\cdots z_{r}=0,{\mbox{ for }}r\leq n.}$
Then, the elements of the form
${\displaystyle {\frac {dz_{i_{1}}}{z_{i_{1}}}}\wedge {\frac {dz_{i_{2}}}{z_{i_{2}}}}\wedge \cdots {\frac {dz_{i_{l}}}{z_{i_{l}}}}\wedge dz_{j_{1}}\wedge \cdots dz_{j_{m}},}$
for ${\displaystyle l+m=k}$, ${\displaystyle i_{\bullet }\leq r}$, ${\displaystyle j_{\bullet }>r}$, form a basis of ${\displaystyle \Omega _{X}^{k}(\operatorname {log} D)|_{V}}$.

Theorem 11.2

${\displaystyle H^{k}(U,\mathbb {C} )=\mathbb {H} ^{k}(X,\Omega _{X}^{\bullet }(\operatorname {log} D)).}$

Hence, we can consider the filtration ${\displaystyle F^{p}H^{k}(U,\mathbb {C} )}$ induced by the naive

${\displaystyle F^{p}\Omega _{X}^{\bullet }(\operatorname {log} D):=q\Omega _{X}^{\geq p}(\operatorname {log} D).}$
Such a filtration induces a pure Hodge structure of weight ${\displaystyle i}$ on
${\displaystyle \operatorname {Gr_{i}^{W}} H^{k}(U)={\begin{cases}0&ik+1.\end{cases}}}$

Remark 11.1

In the case of ${\displaystyle X}$ singular, we take a smooth resolution ${\displaystyle {\hat {X}}}$ with exceptional set ${\displaystyle E=\cup E_{i}}$, where ${\displaystyle E_{i}}$ are simple normal crossing divisors. We iterate the procedure briefly explained in the previous section in a similar manner. Moreover, if ${\displaystyle f:{\bar {X}}\to X}$ is proper and surjective, then

${\displaystyle W^{k-1}H^{k}(X)=\operatorname {ker} f^{*}.}$